A Robust Numerical Algorithm for Computing Maxwell's Transmission Eigenvalue Problems

Abstract

We study a robust and efficient eigensolver for computing a few smallest positive eigenvalues of the three-dimensional Maxwell's transmission eigenvalue problem. The discretized governing equations by the Nédélec edge element result in a large-scale quadratic eigenvalue problem (QEP) for which the spectrum contains many zero eigenvalues and the coefficient matrices consist of patterns in the matrix form XY −1 Z, both of which prevent existing eigenvalue solvers from being efficient. To remedy these difficulties, we rewrite the QEP as a particular nonlinear eigenvalue problem and develop a secant-type iteration, together with an indefinite locally optimal block preconditioned conjugate gradient (LOBPCG) method, to sequentially compute the desired positive eigenvalues. Furthermore, we propose a novel method to solve the linear systems in each iteration of LOBPCG.Intensive numerical experiments show that our proposed method is robust, although the desired real eigenvalues are surrounded by complex eigenvalues.Recently, there have been some papers [12,15,18,19,21,25,28,32,33] addressing numerical computations in transmission eigenvalue problems. In [12], three finite element methods (FEMs) were proposed for solving the two-dimensional (2D) transmission eigenvalue problem. A coupled boundary element method and FEM was introduced for the interior transmission problem in [15]. Then, Sun [32] proposed two iterative methods together with convergence analysis based on the existence theory of the fourth-order reformulation for the transmission eigenvalues [9,30]. A mixed FEM for 2D transmission eigenvalue problems was proposed in [18] and the corresponding non-Hermitian quadratic eigenvalue problem (QEP) was solved by the classical secant iteration with an adaptive Arnoldi method. In [19], Ji, Sun, and Xie used the multilevel correction method to transform the solution of the transmission problem into a series of solutions corresponding to linear boundary value problems and solved them by the multigrid method. Li et al. in [25] rewrote the QEP as a particular parameterized generalized eigenvalue problem (GEP) for which the eigenvalue curves are arranged in a monotonic order so that the desired curves can be sequentially solved with a new secant-type iteration.For a three-dimensional (3D) Maxwell's transmission eigenvalue problem, two FEMs with an adaptive Arnoldi method were proposed in [28]. The resulting GEPs are large, sparse, and non-Hermitian. The numerical challenges for solving the corresponding GEPs are (i) a few of the smallest positive eigenvalues, which may be surrounded by complex eigenvalues, are of interest; (ii) the number of zero eigenvalues of the GEP is huge because the nullity of the discrete double curl operator equals the number of edges in the spanning tree of a finite element mesh [2]; (iii) efficient solution of the associated large sparse linear system in each iteration of the eigensolver. To tackle drawbacks (i) and (ii), in [33], a mixed FEM was applied to an equivalent quadcurl eigenvalue proble...

An efficient numerical algorithm for computing densely distributed positive interior transmission eigenvalues

“…It has been shown [13] that the GEP (13) can be reduced to the QEP as in (15) and (16) with x = u − v in which all nonphysical zero are removed.…”

“…The QEP above can be rewritten as a particular parametrized symmetric definite generalized eigenvalue problem (GEP). For such GEP, the eigenvalue curves can be arranged in a monotonic order so that the desired curves are sequentially located by a new secant-type iteration (see [25] for 2D TEP and [13] for 3D TEP, respectively).…”

A simple low-degree optimal finite element scheme for the elastic transmission eigenvalue problem

“…There are different types of transmission eigenvalue problems, such as the acoustic transmission eigenvalue problem, the electromagnetic transmission eigenvalue problem, and the elastic transmission eigenvalue problem, etc. Since 2010, effective numerical methods for the acoustic transmission eigenvalues have been developed by many researchers [1,8,10,14,15,19,20,23,24,26,27,30,33,[37][38][39], while there are much fewer works for the electromagnetic transmission eigenvalue problem and the elastic transmission eigenvalue problem [18,21,28,32,36,40]. In this paper, we try to develop effective numerical methods for transmission eigenvalue problem of elastic waves.…”

A lowest order mixed finite element method for the elastic transmission eigenvalue problem

“…Since 2010, effective numerical methods for the acoustic transmission eigenvalues have been developed by many researchers [1, 7, 8, 12, 13, 16, 17, 19-21, 23, 28, 31, 33, 34]. There are also much fewer works for the electromagnetic transmission eigenvalue problem [15,26,30]. The goal of this paper is to develop effective numerical methods for transmission eigenvalue problem of elastic waves.…”